Equation of Wave with Constant Velocity/Corollary

Theorem

Let $\phi$ be a wave which is propagated along the $x$-axis in the negative direction with constant velocity $c$ and without change of shape.

Let $\paren {\map \phi x}_{t \mathop = 0} = \map f x$ be the wave profile of $\phi$.

Then the disturbance of $\phi$ at point $x$ and time $t$ can be expressed using the equation:

$\map \phi {x, t} = \map f {x + c t}$

where:

$x$ denotes the distance from the origin along the $x$-axis

$t$ denotes the time.

Proof

We have by hypothesis that the velocity of $\phi$ in the negative direction is $c$.

Hence the velocity of $\phi$ in the positive direction is $-c$.

By Equation of Wave with Constant Velocity:

$\phi = \map f {x - \paren {-c} t}$

Hence the result.

$\blacksquare$

Sources

• 1955: C.A. Coulson: Waves (7th ed.) ... (previous) ... (next): Chapter $\text {I}$: The Equation of Wave Motion: $\S 2$: $(2)$